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teaching and learning of simultaneous equations

Discussion in 'Mathematics' started by ahmed_j_1988, May 21, 2012.

  1. I am carrying out a research project on the teaching and learning of simultaneous equations. I am trying to ascertain what impact the teaching and learning that takes place at ks4 has on his/her a-level maths studies.

    Please do not take too long answering the questions as it would be unfair to you guys! Thanks for all the anticipated help!

    (1) ?How do you find teaching simultaneous equations??
    (2) What sequence do you follow when teaching simultaneous equations, in relation to the different methods available to solve simultaneous equations? Please could you expand by giving examples?
    (3) Do you feel one method is more effective than the other? Please could you answer this in relation to:
    - Making progress in class and in staying in line with the syllabus.
    - Enabling students to answer exam styled question.
    - Preparing students for further mathematical studies i.e. A-levels

    (4)Do you have any other comments on the topic?
  2. Simultaneous equations can be a little algorithmic to teach. I like to introduce the concept with the classic hens and rabbit problem (quoted in Polya's Mathematical Discovery) which is basic but drives home that there is a point to simulatneous equations.
    I teach therefore for understanding allowing leaners to develop their thinking through sucessive iterations i.e. trial and error before teaching the procedures you can use. I think sucessive iterations must be taught first and encouraged to make sne of the algorithms which follow.
    Lastly I like to give students the real application in which simultaneous equations are use in the real world, for flight paths, mobile phone triangulation etc anything to stimulate interest in the topic which at a basic level is very dry.
    Hope this helps.
  3. I am an NQT and taught simultaneous equations for the first time the other day. it was quite daunting, but like with most things, i tend to teach it myself again first. recalling the step by step process. i linked it in with solving equations with one unknown then presented the problem with two, the pupils seemed baffled. Which is when i introduced the step by step method. Having said this, it was great to see some of the maths they 'made up' to try and solve them before i presented the method.

    i did notice however, that they could follow the method when we did it as a group, but when set onto a task of their own they were letting simple things trip them up, like -4x+3x, getting confused with simple negatives.

    I did give them a practical application of a sweet shop, which directly got their attention!

    From reading your post i can see how teaching in an algorithmic way would help. Found it useful. thanks
  4. PaulDG

    PaulDG Occasional commenter

    This is what I hope the ACME report is getting at.

    There are two issues here and it's important to differentiate between them.

    First of all, we all make simple slips - we all sometimes drop a negative sign (I know I do - ask my further maths classes!), write cos 30 = 1/2 when we meant sin 30, etc.

    But it's likely you've found kids who actually don't know what happens if you multiply a negative by a positive, or what happens if you subtract a negative or add to one.

    Essentially, we shouldn't be teaching simultaneous equations to kids who don't "get" directed number and our headlong rush through schemes of work in order to hit short term targets is the cause of this.

    Long Live the Revolution!
  5. Guish

    Guish New commenter

    How would you implement such an approach in a classroom though? If the student is not mature enough for simultaneous equations, he/she should be doing something else/more basic. How would you look after that kid in the same room while others already master simultaneous equations and want to explore some application of it now? A problem may rise in the long run where enough material has not been able to be done because of ability gap among students.

    Anyway, do you have a link to the ACME report? I'd love to read it.

  6. PaulDG

    PaulDG Occasional commenter

    Sets. Maybe vertical ones.

    It's the first link on this list (and it's only really concerned with the top 30% (i.e. the ones who would have gone to a grammar school not all that long ago..) but I think we can imply a lot for lower performers too).
  7. Really?
  8. G'day Paul yeah the problem will always exist in middle to lower streamed classes as we have at Westfields here in Sydney Australia, however I get around it by first giving examples of the three methods Substitution,Elimination and Graphical with plenty of easy examples at first.Students who struggle with the first two methods can do the Graphical method for easier problems and seem to get some insight. Relevance with any maths topic for these students is the key.
  9. PaulDG

    PaulDG Occasional commenter


    Just to say I'm not completely stupid - I also differentiate the work I give to groups that aren't really up to solving simultaneous equations too as, like everyone else, I need kids to pick up a few marks in GCSE and, if the question they get on the day is simple enough, who's to say they won't remember to follow the algorithm and get it right?

    So, like most in England, I expect, I teach the simplest form of elimination to those who need it as an entirely mechanical process.

    Substitution is great for top sets and those likely to do A level - in fact, you often have to teach them elimination as they work out substitution on their own.

    Graphical, well now, that's tricky.

    While we can do graphical solutions using Autograph or Geogebra in the ICT suites, only those who actually can grasp elimination or substitution really have a chance with graphical methods - those who can't solve simultaneous equations can't "see" the equation as the graph (actually, a lot of kids who can really do algebra can't make the connection to the graph either - they often see that as a distraction, something we go over time and time again at A level!).

    And anyway, those who don't "get" directed number are unlikely to be able to properly fill in a table of values and plot the results without computer help..
    Oh absolutely - but relevance of simultaneous equations is an incredibly tricky one for "kids who don't like maths". Kids who like it don't really care about relevance though if they think about it, they can usually come up with applications like cost comparisons. But those who aren't interested aren't interested in things like cost comparisons either...
  10. Guish

    Guish New commenter

    Maybe we should encourage experimentation with any graphing software. This idea can be tough to implement if they're struggling with solving but It'll surely help once solving is mastered. A lot of students panic when they are asked for a geometric interpretation after solving a pair of simultaneous equations(SES). It's just because they have focussed on solving only and probably didn't listen to the teacher when the latter talked about the significance of SES. Give them some pair of equations to solve manually and ask them to use the graphing software to plot the curves as well. They should be able to understand that the solutions represent the pts of intersections if a project is done on that. Transformations of graphs is best understood if students can change parameters/variables in an equation and look at the result of each change instantly. I taught students the transformation of graphs by making them experiment on a graphing software, It was a lot easier for them to understand.

  11. I'd agree that graphing software brings transformations of graphs alive, I'm not convinced this would help with the lack of understanding that Paul is talking about.
    I've tried moving away from graphing software for teaching both y = mx + c and simultaneous equations. In relation to simultaneous equations, this brough some success, especially when looking at equations like x + y = 5.
    I began by getting students to list pairs of numbers that added up to make 5, most stuck with whole numbers but some moved to decimals, we made the link with the equation and coordinates and plotted the points on a graph. We then talked about the 'holes' in our graph and about how we could fill them. The level of understanding wouldn't have been possible to obtain using graphing software.
    I have had a similar experience to Paul in that most don't get the link between x + y = 5 and its graph, working manually helped with that and also getting across the idea that there were lots of 'solutions' to x + y = 5 but a unique one when we also had to satisfy y = x + 1 (which we plotted in a similar way).
    Don't get me wrong, I love ICT but on this occasion, pencil and paper slowed down the process and worked much better.
  12. Guish

    Guish New commenter

    It makes perfect sense when dealing with a linear pair of equations. However, I was thinking about the solving of quadratic equations and linear ones. It'd be tedious to draw and make them understand that the points of intersections are the solutions. You can do it at most twice but more than that would be a loss of time. I agree that they should be able to do it manually. I've no doubt on that. However, giving a series of complex pair of equations and making them graph these equations was the idea. I'm probably talking about an Additional Mathematics lesson or a Form 6 lesson while you are thinking about introducing simultaneous equations to a group of pupils.

    In the same way, when introducing the concept of discriminant to students, we can have a starter exercise where they are given pairs of lines and curves. Each of the examples would give either a tangent to a curve, a line cutting a curve at one point or the line not cutting the curve. We could make them work out the discriminant manually but they graph the equations using the software. Thus, they'll be able to investigate the connection of roots to discriminant. In the cases I'm suggesting, I'm assuming that they understand the basic notion of point of intersection first.

  13. Guish

    Guish New commenter

  14. Hens and rabbits never fail. If you're in a mood to make them laugh as well though, try substituting that with singing the beginning of Old McDonld had a farm up to the ducks part and then introduce some sheep as well :p


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